Theorem univ 4544

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3866	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3877	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4282	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2232	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1375  Vcvv 2779  𝒫 cpw 3629  ∪ cuni 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-uni 3868

This theorem is referenced by: (None)